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Theorem prnz 4743
Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
Hypothesis
Ref Expression
prnz.1 𝐴 ∈ V
Assertion
Ref Expression
prnz {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz
StepHypRef Expression
1 prnz.1 . . 3 𝐴 ∈ V
21prid1 4728 . 2 𝐴 ∈ {𝐴, 𝐵}
32ne0ii 4302 1 {𝐴, 𝐵} ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wne 2944  Vcvv 3448  c0 4287  {cpr 4593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-v 3450  df-dif 3918  df-un 3920  df-nul 4288  df-sn 4592  df-pr 4594
This theorem is referenced by:  opnz  5435  propssopi  5470  fiint  9275  wilthlem2  26434  upgrbi  28086  wlkvtxiedg  28615  shincli  30346  chincli  30444  spr0nelg  45742  sprvalpwn0  45749
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